Projections of the sphere graph to the arc graph of a surface

TL;DR

This paper constructs a canonical Lipschitz retraction from the sphere graph of a doubled handlebody to the arc graph of a surface, linking their geometric structures and providing bounds on the projection.

Contribution

It introduces a new canonical Lipschitz retraction from the sphere graph to the arc graph, establishing a geometric connection between these complexes.

Findings

The retraction is uniformly Lipschitz.

It is within a bounded distance of the nearest point projection.

The construction applies to maps inducing isomorphisms of fundamental groups.

Abstract

Let S be a compact surface, and M be the double of a handlebody. Given a homotopy class of maps from S to M inducing an isomorphism of fundamental groups, we describe a canonical uniformly lipschitz retraction of the sphere graph of M to the arc graph of S. We also show that this retraction is a uniformly bounded distance from the nearest point projection map.